Éva Orbán-Mihálykó

Intermediate storage in batch/continuous processing systems under stochastic operation conditions

A mathematical model and model-based method is presented to design the intermediate storages aiming to buffer the operational differences between the batch and continuous subsystems in processing systems. The occurrence times of the inputs are assumed to be described by a Poisson process, while the amounts of the material transferred by the batch units allowed changing according to general probability distributions. Based on the stochastic differential equation model of operation, integral equations for determining the overflow and underflow probabilities of a finite storage are formulated for both infinite and finite operation horizons that provide the basis for the rational design of such intermediate storages. Analytical solutions to the integral equations for infinite horizons are derived in the cases of constant and exponentially distributed inputs. For the batch sizes described by general distribution functions, solutions to the integral equations are obtained in the form of approximating functions generated by stochastic simulation. A number of numerical experiments with exponential, normal and lognormal distributions of the batch sizes are presented and analyzed. The effects of process parameters on the design are also investigated.

Sizing of pipeline capacities in processing systems under stochastic operation conditions

Abstract Mathematical models and model-based methods are presented for reliability-based design of the capacity of pipelines aiming to transfer material or energy in batches under stochastic operation conditions. The occurrence times of the transfers are assumed to be described by Poisson processes, while the duration of the material and energy transfers may be constant or may vary randomly according to general probability distributions. Approximating analytical solutions are derived for constant duration batches, developed in terms of scan statistics, based on which two heuristics are formulated for sizing the pipeline capacities for constant batches. In the case of transferring random distribution batches described by general distribution functions, solutions are obtained by means of stochastic simulation based on an algorithm developed for that purpose. It is shown that the pipeline with random duration of batches becomes oversized using the heuristics of constant batches the extent of which depends on the standard deviation of the corresponding distribution. Combining the approximating formulae and the algorithm, a heuristic is formulated for sizing the pipeline also for random batches. A number of simulation experiences with exponential, normal, log normal and uniform distributions of the transfer sizes are presented and analyzed. The method works satisfactorily even in the case of bimodal distribution of batch sizes. Examples are presented to illustrate the proposed methodology.

A generalization of the Thurstone method for multiple choice and incomplete paired comparisons

A ranking method based on paired comparisons is proposed. The object’s characteristics are considered as random variables and the observers judge about their differences. The differences are classified. More than two classes are allowed. Assuming Gauss distributed latent random variables we set up the likelihood function and estimate the parameters by the maximum likelihood method. The rank of the objects is the order of the expectations. We analyse the log-likelihood function and provide reasonable conditions for the existence of the maximum value and the uniqueness of the maximizer. Some illustrative examples are also presented. The method can be applied in case of incomplete comparisons as well. It allows constructing confidence intervals for the probabilities and testing the hypothesis that there are no significant differences between the expectations.

Sizing intermediate storage with stochastic equipment failures under general operation conditions

Abstract An algorithm and simulation program have been developed for sizing intermediate storages of batch/semicontinuous systems taking into account stochastic equipment failures under general operation conditions. The method is based on the observation that any process of this system can be built up from a random sequence of failure cycles of finite number. By means of simulation and statistical evaluation of the results of the simulation runs the storage is sized at a given significance level. The computation time is favourable, and it appears to be a linear function of the number of the failure cycles.

Incomplete paired comparisons in case of multiple choice and general log-concave probability density functions

A scoring method based on paired comparison allowing multiple choice is investigated. We allow general log-concave probability density functions for the random variables describing the difference of the objects. This case involves Bradley–Terry models and Thurstone models as well. A sufficient condition is proved for the existence and uniqueness of the maximum likelihood estimation of the parameters in case of incomplete comparisons. The axiomatic properties of the method are also investigated.

Application of Advanced Integrodifferential Equations in Insurance Mathematics and Process Engineering

In this paper we consider a dual risk model with general inter-arrival time distribution and general size distribution. A special Gerber–Shiu discounted penalty function is defined and an integral equation is derived for it in the case of dependent inter-arrival times and sizes. We prove the existence and uniqueness of the solution of the integral equation in the set of bounded functions and we show that the solution tends to zero exponentially. If the density function of the inter-arrival time satisfies a linear differential equation with constant coefficients, the integral equation is transformed into an integrodifferential equation with advances in arguments and an explicit solution is given without any assumption on the size distribution. We also present a link between the Lundberg fundamental equations of the Sparre Andersen risk model and the dual risk model.